A136405 Triangle read by rows: T(n,k) is the number of bi-partitions of the pair (n,k) into pairs (n_i,k_i) of positive integers such that sum k_i = k and sum n_i*k_i = n.

1, 1, 2, 1, 1, 3, 1, 3, 2, 5, 1, 2, 4, 3, 7, 1, 4, 6, 7, 5, 11, 1, 3, 7, 8, 11, 7, 15, 1, 5, 8, 16, 14, 17, 11, 22, 1, 4, 12, 14, 23, 21, 25, 15, 30, 1, 6, 12, 24, 29, 38, 33, 37, 22, 42, 1, 5, 15, 24, 41, 42, 57, 47, 52, 30, 56, 1, 7, 18, 37, 47, 75, 68, 87, 70, 74, 42, 77

Offset: 1

Author

Benoit Jubin, Apr 13 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 2;
  1, 1,  3;
  1, 3,  2,  5;
  1, 2,  4,  3,  7;
  1, 4,  6,  7,  5, 11;
  1, 3,  7,  8, 11,  7, 15;
  1, 5,  8, 16, 14, 17, 11, 22;
  1, 4, 12, 14, 23, 21, 25, 15, 30;
  1, 6, 12, 24, 29, 38, 33, 37, 22, 42;
  ...
T(4,2) = 3 since (4,2) can be bi-partitioned as (2,2) or ((1,1),(3,1)) or ((2,1),(2,1)).
T(5,3) = 4 since (5,3) can be bi-partitioned as ((1,1),(2,2)) or ((3,1),(1,2)) or ((1,1),(1,1),(3,1)) or ((1,1),(2,1),(2,1)).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums are A006171.

Cf. A000041, A028242, A090806, A136406.

Programs

PARI

P(k, w, n)={prod(i=1, k, 1 - x^(i*w) + O(x*x^(n-k*w)))}
T(n)={Vecrev(polcoef(prod(w=1, n, sum(k=0, n\w, (x*y)^(k*w)/P(k,w,n))), n)/y)}
{ for(n=1, 10, print(T(n))) } \\ Andrew Howroyd, Oct 23 2019

Formula

T(n,1) = 1.

T(n,2) = A028242(n).

T(n,n) = A000041(n).

Extensions

Terms a(57) and beyond from Andrew Howroyd, Oct 23 2019